An error-corrected qubit is a logical qubit that is protected from errors using quantum [error correction] (QEC) techniques. More precisely, error-corrected qubits are highly-entangled quantum states where one or several qubit-worth of information have been encoded globally.
Note that [physical qubits] may already include a partial error-protection mechanism. For instance, the dual-rail qubits used by Quandela enable the detection of a photon loss.
The [threshold theorem]( https://en.wikipedia.org/wiki/Threshold_theorem) states that the overall information can be arbitrarily well protected, provided given that the physical [noise] remains local and below a certain threshold value.
An important class of error-corrected qubits
Error-corrected qubits can be constructed as the common eigenstates of a set of commuting operators, the stabilizer operators. Such construction is widely used in [error-correction]. Here are some important examples:
- Repetition code: it encodes one-qubit worth of information into states |0⟩L=|0⋯0⟩0L=0⋯0and . It can detect up to 50% bit-flip errors (where |0⟩0is exchanged with |1⟩1) but no phase-flip errors (where |+⟩+ is exchanged with |−⟩−). The stabilizer operators are ZiZiwhere is a single qubit Z operator acting on qubit i.
- Surface code: imagine that physical qubits are located on the edges of a two-dimensional grid. For each square p formed by the grid with qubits (a,b,c,d) on the edges we define a so-called “plaquette” operator gp=ZaZbZcZdgp=ZaZbZcZd. For each crossing s in the grid where edges (a,b,c,d) intersect, we define a so-called “star” operator gs=XaXbXcXdgs=XaXbXcXd. The common eigenstates of all the star and plaquette operators form a “surface code” that encodes one (or two if we set periodic boundary conditions on the grid) qubit worth of information.
Frequently asked questions about error-corrected qubits
- How many physical qubits does it take to build one error-corrected qubit? It depends on the specific encoding that we use (the specific stabilizer operators if one is considering stabilizer states), on the noise level, and on several other parameters, but a rough estimate is around 1000 physical qubits for one error-corrected qubit. There is a lot of research to try and bring this number down as much as possible.
- Why do we need error-corrected qubits? Quantum systems are fragile and prone to errors. If the information is stored at the level of physical qubits, it will quickly get corrupted. Error-corrected qubits store information globally and therefore dissociate physical information from logical information, see [error correction] for more details.